cph said:
Would it seem reasonable that so-called ‘hot jupiters’ are in resonance i.e. have a stable orbit?
Resonance is not required for a stable orbit. A large number (and fraction) of multi-planet extrasolar systems are known to be in resonance however. It is a little more complicated with hot-jupiters, because they are so close in. In general, the hot-jupiters with eccentric orbits can be assumed to not be in significant resonances.
cph said:
If one utilized our stellar system as a simulation, with the addition of a hot jupiter, then what would the resonance be?
There would be many possible resonances... i.e. there are many semi-majoraxes at which the hot Jupiter could be in a resonance with our-own jupiter.
cph said:
Would it be calculated as non integral?
What do you mean?---Resonances are defined as (near) integral.
cph said:
Then if our cold gas giant were discarded, would this then seem to change the resonance.
Well, if there's nothing to resonate with... then yes, it would change the resonance.
It could still be in resonance with saturn (more accurately, saturn could be in resonance with it).
cph said:
Hence would one have the prediction of no cold gas giants for any of the hot Jupiter systems?
There are a tremendous number of 'if's and steps in that argument... many of which I don't quite follow.
Overall: if the migration theory is correct, it is reasonable to assume that other cold-giants might have been ejected or otherwise destroyed. It is also possible that other bodies could survive. Additionally, if some hot-jupiters managed to form
in situ, in might be a good method of seeding other giants further out (via resonance perturbations).